# Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that

• $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$.

• Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{Z}/(a^{2}+b^{2})\mathbb{Z}$, if $a,b$ are relatively prime.

My attempt was to define a map, $\varphi:\mathbb{Z}[i] \to \mathbb{Z}/10\mathbb{Z}$ and show that the kernel is the ideal generated by $\langle{3-i\rangle}$. But I couldn’t think of such a map. Anyhow, any ideas would be helpful.

This diagram shows the Gaussian integers modulo $3-i$.
The red points shown are all considered to be $0$ but their locations in $\mathbb Z[i]$ are $0$, $3-i$, $i(3-i)$ and $3-i + i(3-i)$. Every congruence class must be inside that box once and you can see there are $10$ of them.
The arrows show adding by $1$ each time. Doing that takes you through every equivalence class and then back to the start.
So $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$.